I think time invested into studying real analysis pays off because then you can later study measure theory, functional analysis and more advanced probability to deal with curse of dimensionality and whatnot.
edit: I started studying the book linked above starting from chapter 4 since the first 3 chapters are familiar from discrete math. Then did chapter 5, skimmed chapters 6(little linear algebra), 7, 8 (most "transition to higher math" books contain this stuff) and am currently in chapter 9.
My previous college calculus was far from the rigour of Rudin but also far from a cookbook flavour. Unless by 'cookbook' you mean the chain rule and differentials of standard forms. It's not that hard to teach this stuff at least it's no harder than, say, geometry. I found trigonometry far more difficult.
We were first taught limits then came differentiation of polynomials using infitesimals.Chain rule was introduced. Then we ventured into differentiation of other functions. Integration was first introduced much like riemann integrals then came integration as an inverse of differentiation.
http://www.cs.cornell.edu/jeh/book%20June%2014,%202017pdf.pd... (Foundations of Data Science by Bloom/Hopcroft/Kannan)